**Augmenting the great icosidodecahedron**

The great icosidodecahedron, vertex figure (^{5}/_{2},3,^{5}/_{2},3)

Left: Each
triangular face of the great icosidodecahedron can be augmented with a triangular
pyramid or tetrahedron. Each 3 in the vertex figure is replaced by
(3,3) The original vertex figure becomes (^{5}/_{2},__3,3__,^{5}/_{2},__3,3__)
with new vertices of (3,3,3). All
vertices are on the exterior of the polyhedron.

Right: The above figure is shown with the tetrahedra shown in framework style. A framework of this model is linked here.

Left:
Another augmentation of the great icosidodecahedron can be
obtained by excavating each pentagrammic face with a pentagrammic
pyramid completing a cycle around the axis and each triangular face with a triangular
pyramid or tetrahedron which again completes a cycle around the axis. Each ^{5}/_{2}
and each 3 in the vertex figure is replaced by
(3,3) The original vertex figure becomes (__3,3__,__3,3__,__3,3__,__3,3__)/3
or (3^{8})/3 with new vertices of (3,3,3,3,3)/2 and (3,3,3)
respectively. The faces of the pentagrammic pyramids are shown in yellow and
the tetrahedral faces in blue. All
vertices lie on the convex hull.

Centre:
A further augmentation of the great icosidodecahedron can be
obtained by excavating each pentagrammic face with a pentagrammic
antiprism which completing a cycle around the axis and each triangular face with a triangular
pyramid or tetrahedron which again completes a cycle around the axis. Each ^{5}/_{2}
in the vertex figure is replaced by a (3,3,3) and each 3 in the vertex figure is replaced by
(3,3) The original vertex figure becomes (__3,3,3__,__3,3__,__3,3,3__,__3,3__)/3
or (3^{10})/3
with new vertices of (^{5}/_{2},3,3,3) and (3,3,3) respectively.
The triangular faces of the pentagrammic antiprisms are shown in yellow and the
tetrahedral faces in blue. All
vertices are on the exterior of the polyhedron.

Right:
A further
augmentation of the great icosidodecahedron can be obtained as follows:
Excavate one pentagrammic cap of a crossed pentagrammic
antiprism with a pentagrammic pyramid. The
resulting polyhedron is equivalent to a diminished
great icosahedron. This polyhedron is not locally convex but the act of joining
it to the great icosidodecahedron removes the retrograde polygon . Excavate
each of the pentagrammic faces of the great icosidodecahedron
with a diminished great icosahedron.
The ^{5}/_{2}
in the vertex figure is replaced by a (3,3,3)/2 The original vertex figures become (__3,3,3__,3,__3,3,3__,3)/3 - or (3^{8})/3 with new vertices
from the diminished great icosahedron of (3,3,3,3,3)/2 - or (3^{5})/2. The original triangular faces
of the stellatruncated cube are shown in blue, those of the anti-prisms in
yellow and the pyramids red. All
vertices are on the exterior of the polyhedron.